Question

Find a unit vector u in the same direction as v= (8,-4)

Answer 1

Explanation:

In this exercise we have the following vector:

$\vec{v}=(8,-4)$

So we need to find a unit vector $\vec{u}$ which is defined as follows:

$\vec{u}=\frac{\vec{v}}{\mid \vec{v} \mid}$

Finding the magnitude of the vector:

$\mid \vec{v} \mid=\sqrt{(8)^2+(-4)^2} \\ \\ \mid \vec{v} \mid=\sqrt{64+16} \\ \\ \mid \vec{v} \mid=\sqrt{80} \\ \\ \mid \vec{v} \mid=4\sqrt{5}$

Then, by substituting the unit vector is:

$\vec{u}=\frac{(8,-4)}{4\sqrt{5}} \\ \\ \vec{u}=\frac{(8,-4)}{4\sqrt{5}} \\ \\ \vec{u}=(\frac{8}{4\sqrt{5}},-\frac{4}{4\sqrt{5}})$

$\vec{u}=(\frac{8}{4\sqrt{5}}\frac{\sqrt{5}}{\sqrt{5}},-\frac{4}{4\sqrt{5}}\frac{\sqrt{5}}{\sqrt{5}}) \\ \\ \\ \boxed{\vec{u}=\left(\frac{2\sqrt{5}}{5},-\frac{5}{\sqrt{5}}\right)}$